Question

Given odd positive integers \( x, y, z \), which of the following is not necessarily true?

• A. \( x^2 y^2 z^2 \) is odd
• B. \( 3(x^2 + y^3)z^2 \) is even
• C. \( 5x + y + z^4 \) is odd
• D. \( \frac{z^2(x^4 + y^4)}{2} \) is even

Hint:

Odd + odd = even, but dividing even by 2 doesn’t guarantee even result—it may become odd.

Answer 1

The Correct Option is D

All variables are odd integers.
Option (a): Each of \( x^2, y^2, z^2 \) is odd (odd squared is odd). \[ \text{Product of three odd numbers is odd.} \quad \boxed{\text{Always true}} \] Option (b): Odd squared and odd cubed → still odd. So, \( x^2 + y^3 = \text{odd + odd} = \text{even} \)
Even × odd \( z^2 \) = even, then × 3 = still even. \[ \boxed{\text{Always true}} \] Option (c): Sum of three odd terms \( 5x + y + z^4 \): Odd + odd + odd = odd \[ \boxed{\text{Always true}} \] Option (d): Check: \( x^4 + y^4 = \text{odd + odd} = \text{even} \), and \( z^2 \) is odd.
Now multiply: even × odd = even
Now divide by 2: Might be integer, might not — depends if even is divisible by 2.
Example: \( x = 1, y = 1, z = 1 x^4 + y^4 = 1 + 1 = 2 \) Then \( z^2(x^4 + y^4)/2 = 1 \times 2 / 2 = 1 \) — odd! So, the result is not always even. \[ \boxed{\text{Not necessarily true}} \] \[ \boxed{(d)} \text{ is the correct answer} \]